The other direction
Two angles total 180°. Must they be next to each other on a line?
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Step 1 of 4
- One angle is 80°. It is drawn alone.
- Another is 100°. It is drawn far away.
- 80° + 100° = 180°, but they share no line.
- The same two angles, now a real linear pair.
The rule
No. Two angles can total 180° and be far apart.
A converse turns a claim around. The theorem says: linear pair, so 180°. The converse says: 180°, so linear pair.
That converse is false. The angles above total 180°. They share no vertex and no arm. One such drawing shows the claim is false.
Axiom. An axiom is a fact we accept without proof. Suppose two angles share a vertex and an arm. They lie on opposite sides of that arm. If the angles total 180°, their outer arms form a straight line.
| Claim | True? | Reason |
|---|---|---|
| Any two angles that total 180° sit on a line. | No | 80° and 100°, drawn apart. |
| Two angles share a vertex and an arm, lie on opposite sides of it, and total 180°. Their outer arms form a line. | Yes | The axiom. |